Investigating the maths of Deal Or No Deal

Feature Alex Carter 21 May 2014 - 07:00

Step inside the dream factory for a chat about Noel Edmonds' beard and probability theory...

My first memory is of my dad getting into a fight with Noel Edmonds on the set of Telly Addicts. He had gotten so bored with the endless trivia that he fell asleep during one question, and woke up during another, and ran onto the stage to complain that David Jason wasn’t The Fugitive. If he found a colourful 80s gameshow boring, good lord knows what he makes of random guesstheboxathon Deal Or No Deal. He would probably implode into a coma.

But is it really just a game of chance? Unfortunately, try as you might, you can never get away from that agonising itch that it’s all just a completely random number guessing game. Albeit one with a delusional man who has phone conversations with a non-existent financial warlord, and 4000 gallons of tears. Delve deep into the mathematics though and instead of finding some formula, it turns out it’s even more down to probabilities and statistics than you thought.

Let’s start with the basics. You have a 1/22 chance of picking the biggest prize at the start (although equally you have a 1/22 chance of picking any value).  If you consider success to be one of the five biggest, it’s a 5/22 chance. OK, let’s try to be clever. What’s the chance that you won’t pick the highest box on each turn? Assuming the first box you pick (to keep) isn’t the £250,000 prize, the chance of continually avoiding the star prize decreases as the boxes are removed. Taking the product gives us…

Damn. There’s no way of getting around it, the probability of “winning” Deal or no Deal is 5/22, about 23%. This is, incidentally, the percentage of Noel Edmonds’ face that is beard.

There are however some twists to try and obfuscate the chance. The banker, and the last box.

Every game begins the same way. Someone is chosen at “random” from the box keepers to play. The amount in their box is what they win, unless they swap at the end. My first thought, as I’m sure was yours, was to think of former Miss Earth’s Highest IQ, Marilyn vos Savant.

The Monty Hall Problem she postulated the answer for is a counter-intuitive (that’s maths speak for “no one really understands it unless they have a calculator and a beard”) probability trick loosely based on the game show Let’s Make A Deal. You’re probably lying if you say you remember it, because it was never shown here. In this thought experiment, you are faced with three doors, behind one of which is a car. The other two, goats. You pick a door but don’t open it, and then the host opens one of the other doors, to reveal a shiny new goat. The problem is this: do you stick, or twist? If you said “it doesn’t matter you idiot, there’s only two doors left so it’s 1/2”, then surprisingly you’d be wrong.

Since the probability of you picking a winner first time is only 1/3, the probability either of the other doors is a winner is 2/3. But even though the host has revealed a non-winning door, it doesn’t change that your door has only a 1/3 probability of having the car; the host opening door simply guarantees that the door he didn’t open has the 2/3 chance all to itself. Therefore you should switch doors. Of course, the real answer is you should take the goat, because as Goat Simulator has shown us, goats can provide tremendous entertainment.

On the face of it, then, you should switch boxes at the end of Deal Or No Deal, because you’ve acquired new information throughout the game by eliminating the boxes. Unfortunately, this doesn’t apply – Edmonds himself doesn’t ever reveal anything about the boxes you haven’t opened. The closest he ever comes to giving any information is via the banker, and all he ever does is make an offer based on the value remaining. Without extra information on the boxes you haven’t opened, the odds of getting the swap right are 50/50, just like you’d expect. So back to square one then. 

After the first box is chosen, our contestant picks boxes at random to throw away, with each one comes an outpouring of grief and tears normally reserved for an X Factor sob story. A number is flashed up on screen, wonderfully colour coded so we know which are the small ones and which are the big ones without having to think too hard. Then another random box is chosen, and more tears. This continues until there are no more boxes left or someone cries themselves into dehydration.

Of course, this was the way the game was designed. In fact, the Gambling Commission met with the producers a few years ago because they suspected that there was in fact no skill involved and was therefore a lottery. Lotteries are banned in this country with one obvious exception, so Noel came close to being thrown back into television obscurity. This rule, incidentally, is why daytime TV shows run stupid quizzes with questions like “Paris is the capital of which country?  A: France, B: Tom Cruise or C: Kwik Save”, because apparently this involves some sort of skill. Much in the same way as breathing, or not falling over. Of course, the show is still on the air – the producers didn’t even attempt to prove it’s not a lottery, they just pointed out it can’t be gambling if the contestants don’t pay to enter.

So is there any way to get one over on them? Perhaps. Noel Edmonds and his imaginary phone calls.

The banker turns up at the end of each round of box opening via telephone to hurl insults and then, crucially, to offer the contestant a deal, based on the current values left on the board. This is where it becomes a game of skill, albeit a fairly simple one if you’re familiar with game theory. The banker makes an offer based on the root mean square (RMS) value v of the remaining boxes, which involves squaring the values, taking the mean, and then square rooting again.

Basically it’s a fancy term for a middle weighted average, so that any of the extreme values don’t skew the average too much. The banker tends to offer an approximation of this, based on how far along the game is and whether the player is “playing well” i.e. if they’re providing some decent entertainment value like crying profusely or wearing a loud shirt. If it’s early in the game or you’re doing things like whipping a calculator out and explaining what a root mean square is, he’ll probably offer you a couple of paper clips and a slap in the thorax. Thus, the strategy is to provide enough entertainment value by pretending to think people have any control of any of the boxes, while thanking everyone on the planet and crying. This still relies on chance of course - you need to open a few blues boxes to weight the average higher or else any deal will be rubbish.

  

Either way, the ultimate decision on whether to deal is whether or not you’re “beating the mean”. If you’re offered substantially lower than the RMS of the remaining boxes, tell the banker to bank off. If he offers more, you should probably take it, especially if you’ve only got a few reds left. If you’ve only got one red left, don’t get greedy, just take the offer, even if you think it’s too low. The best you can hope is that you’ve been enough of a loud crying sadsack that he takes pity on you and gives you a good offer. Fortunately for the rest of us, the set of crying sadsack doesn’t overlap well with mathematicians, so “entertainment” ensues.

So what counts as a success?  Well, the expected value of the boxes before the game is given by the sum of the values, weighted by the chance of finding them.

“But that’s just the average of all the prizes!” I hear you yell, and of course you’re right. This works out to about £25,712.12. But since this is still a game of chance, you have to weigh getting this offer versus the chance of getting a higher box. There are only five boxes with higher values than this, and since each box has the same chance, your chance of winning is 5/22, or 23%. Which is what I already told you.

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Slow news day?!

There's no correlation between the amounts of news on the site and the features. Also, a news story takes minutes to write, a piece like this takes hours at least. So no, not a slow news day, but it wouldn't make a jot of difference if it is! - Simon

There is the distinct probably that your use of the word "correlation" was increased by the mathematical nature of the article - in this case an element of causation where the dependent variable "y" (use of a maths term) is affected by the independent variable "x" (preceded by a maths article).

Similarly, the chances of my proposing this response is greatly increased by your own comment, to the extent that a zero value of the previously described variable "y" would almost certainly result in the chances of this comment being zero. Approached from a different angle, you writing such a comment is mutually exclusive to my NOT writing a comment - but this only rings true if we highlight the level of dependency my comment has on yours. Had I NOT posted this comment, yours would still exist unaffected.

It is also highly probable that I have let this comment run away from me a bit, but since much of that particular analysis would be qualitative and highly subjective, it is more difficult to be precise. Let's just all have a cup of tea and agree that yesterday's "Lost River" trailer was really, really weird.

Excellent piece! I love the smell of mathematics in the morning!

You win, we lose!

I enjoyed reading that.

I was only teasing. It was a brilliantly entertaining article.

I have a few family members obsessed with this pap and usually get the micky took out of me for saying it's all just chance. I've always also wondered what exactly the math involved would be (I thought about the Monty Hall thing but didn't know how the banker and number of boxes would skew it) but was too lazy to actually research it.

Long story not quite so long: thanks for an entertaining peek at the numbers involved and for something I can shove in my my family's collective face.

I enjoyed this. I'm no expert on maths but I do find items like this fascinating. Thanks DoG, something a bit different and enjoyable :)

Excellent. Something properly geeky on DOG... Keep it up :)

Wonderful piece Alex, simultaneously educational yet thoroughly entertaining. A similar piece on the psychological/mathematices behind Goldenballs would be appreciated.

I love pieces like this, that let us geek out about the most obscurely un-talked-out stuff, nice one Alex!

Goats scare me.

DoG, your writers make my day.

Do a mathematical equation on the percentage of how likely it is that any of the box openers who are sure that 'it's a blue love' even though there's NO POSSIBLE WAY THEY COULD KNOW THAT are tw@ts.

I'll save you the trouble. It's 100%.

I came here for the beard discussion . . . disappointed much

Id have a 0% chance of winning as after a couple of minutes id have to leave the studio or release all my pent up rage on edmund's smarmy bearded face

This story was actually so much more entertaining than the show itself. More stories in this vein, please.

My colleagues were wondering though why I was sniggering so much behind my computer.

I have along running disagreement with a work colleague who refuses to accept the 1/3, 2/3 odds of the Monty Hall problem, Deal or no deal gets raised a lot in those discussions.

A couple more minutes to check that last equation wasn't wrong would have been good :)
Despite that, great article. Anything mentioning the Monty Hall problem wins my approval.

This show should be canned for crimes against probability.

An interesting question does come out of it though: if you are down to the last two boxes, and they are 1p and £250,000 - what's the lowest banker's offer you would accept?

Ask them how they would feel if there were 100 doors, and Monty revealed 98 goats.

But are the bankers and /or quizmaster aware of what is behind the doors or in the boxes? Because that would significantly change things and make it more of a psychological game. I have seen quizmasters "subtly hinting" to contestants that they should change their door choice, which would suggest at the fact that the show wants to pay out the big prize. What if the bankers are in a generous mood because there haven't been many big payouts for a while? or the other way around?
In the end almost no tv quiz is truly up to chance; there is always the human element how ever tiny.

The monty Hall problem does apply here. It doesn't matter if you know where the money is as there is a condition imposed on the problem, the assumption that the contestant had got to the end without revealing the 250k. At the start you have a 1 In 22 chance of picking the top prize and a 21 in 22 chance of not. By imposing the condition that the 250k had not been found the probability that it is in your box is 1 in 22. You should always swap if given the opportunity

Ok. I'm going to ask if no-one else is, as it's killing me. Is Paris the capital of Tom Cruise?

I came in thinking "why is this on den of geek?" and I left thinking that this is much better than most of Charlie Brooker's recent literary work (as in his columns, not Black Mirror or Newswipe). Well played, sir!

Brilliant feature. My brain hurts but at least there was no mention of Dawson's Creek or Pretty Little Liars.

I agree, the host knowing is only relevant in order to reduce the contestants options from three to two. In deal of no deal this happens automatically so it still applies.

I disagree. The 250k has equal chance of either box. Monty Hall works because the host knowingly reveals an empty door. Here it has not been revealed through chance. Run 10 simulations you'll see.

Neither Noel or The Banker knows whats in the boxes. I promise.

If you're down to three boxes with the 250k remaining (and it's not revealed on the next turn) then surely it must apply?

A revolver to shoot myself with, couldn't face anyone I knew if i'd been on the show.

It'd take much more than quarter of a mill.

Surely upon seeing Edmond's face, we all lose.

Monty hall is set to mess with your head. The part where the host shows an empty door is a mis direct (the host always reveals an empty door and this is the key, the prize is never revealed by the host). If you ignore that step and keep the door in play. The choice is between one door or the two other doors (at least one of which will be empty because there is only one prize). The 2 door choice the better option. Here it is all straight chance. Each box has a 1/22 chance and nothing changes that in the process. If you get down to the last 2 or 3 it's because of chance and each box is still equally likely to have the prize. At the end you are swapping a 1/22 chance box for another. The opening of box 3 could be the prize but in Monty hall it never is.

Like to take your word for it but how do you know that for sure?
In the case of Deal or No deal I think the quizmaster doesn't know, but the bankers could very well be in the know.
Seen a lot quizzes in the past where it was evident that the quizmaster knew exactly which door/box/etc had the big prize and actively steered contestants choices in the direction the show wanted to maximize entertainment value.
Ever seen the movie "Quiz Show"?

Things like the real life events inspiring the film "Quiz Show" are exactly the reason why Deal or No Deal (and other game shows) has an independent adjudicator, separate from the production, who oversees the whole process to ensure it's all fair and above board. Besides, the wealth of statistical data gathered from over 2500 broadcast games throws up absolutely no evidence to suggest anything fishy. (If you don't believe me, feel free to set up a hypothesis test using the data from those 2500 games yourself and confirm it's true.)

Interested to know why the author claims that the "banker makes an offer based on the root mean square (RMS) value v of the remaining boxes". The root-mean-square measure was suggested by fans of the show as a benchmark for what might constitute a fair offer, but to my knowledge has never been part of the Banker's calculations when determining how much to offer. In fact, statistical evidence suggests that there's too much variation around the RMS value for it to have much significance.
It's also worth pointing out that both the other formulae quoted are mathematically incorrect too. I'll leave the proper geeks here to determine why.

Interesting, the only thing I really didn't like about the article was that he was trying so hard to Brooker that it hurt.

Now, the world don't move to the beat of just one drum,
What might be right for you, may not be right for some.
A man is born, he's a man of means.
Then along come two, they got nothing but their jeans.

But they got, Diff'rent Strokes.
It takes, Diff'rent Strokes.
It takes, Diff'rent Strokes to move the world.

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